Pulsatile Flow of a Two-Fluid Model for Blood Flow through Arterial Stenosis

Pulsatile flow of a two-fluid model for blood flow through stenosed narrow arteries is studied through a mathematical analysis. Blood is treated as two-phase fluid model with the suspension of all the erythrocytes in the as Herschel-Bulkley fluid and the plasma in the peripheral layer as a Newtonian fluid. Perturbation method is used to solve the system of nonlinear partial differential equations. The expressions for velocity, wall shear stress, plug core radius, flow rate and resistance to flow are obtained. The variations of these flow quantities with stenosis size, yield stress, axial distance, pulsatility and amplitude are analyzed. It is found that pressure drop, plug core radius, wall shear stress and resistance to flow increase as the yield stress or stenosis size increases while all other parameters held constant. It is observed that the percentage of increase in the magnitudes of the wall shear stress and resistance to flow over the uniform diameter tube is considerably very low for the present two-fluid model compared with that of the single-fluid model of the HerschelBulkley fluid. Thus, the presence of the peripheral layer helps in the functioning of the diseased arterial system.


Introduction
The analysis of blood flow through stenosed arteries is very important because of the fact that the cause and development of many arterial diseases leading to the malfunction of the cardiovascular system are, to a great extent, related to the flow characteristics of blood together with the geometry of the blood vessels.Among the various arterial diseases, the development of arteriosclerosis in blood vessels is quite common which may be attributed to the accumulation of lipids in the arterial wall or pathological changes in the tissue structure 1 .Arteries are narrowed by the development of atherosclerotic plaques that protrude into the lumen, resulting in stenosed arteries.When an obstruction is developed in an artery, one of the most serious consequences is the increased resistance and the associated reduction of the blood flow to the particular vascular bed supplied by the artery.Also, the continual flow of blood may lead to shearing of the superficial layer of the plaques, parts of which may be deposited in some other blood vessel forming thrombus.Thus, the presence of a stenosis can lead to the serious circulatory disorder.
Several theoretical and experimental attempts have been made to study the blood flow characteristics due to the presence of a stenosis in the arterial lumen of a blood vessel 2-10 .It has been reported that the hydrodynamic factors play an important role in the formation of stenosis 11, 12 and hence, the study of the blood flow through a stenosed tube is very important.Many authors have dealt with this problem treating blood as a Newtonian fluid and assuming the flow to be steady 13-16 .Since the blood flow through narrow arteries is highly pulsatile, more attempts have been made to study the pulsatile flow of blood treating blood as a Newtonian fluid 3, 6-8, 17-19 .The Newtonian behavior may be true in larger arteries, but, blood, being a suspension of cells in plasma, exhibits nonNewtonian behavior at low-shear rates γ < 10/scc in small diameter arteries 0.02 mm-0.1 mm ; particularly, in diseased state, the actual flow is distinctly pulsatile 2, 20-25 .Several attempts have been made to study the nonNewtonian behavior and pulsatile flow of blood through stenosed tubes 2, 4, 9, 10, 26-28 .
Bugliarello and Sevilla 29 and Cokelet 30 have shown experimentally that for blood flowing through narrow blood vessels, there is an outer phase peripheral layer of plasma Newtonian fluid and an inner phase core region of suspension of all the erythrocytes as a nonNewtonian fluid.Their experimentally measured velocity profiles in the tubes confirm the impossibility of representing the velocity distribution by a single-phase fluid model which ignores the presence of the peripheral layer outer layer that plays a crucial role in determining the flow patterns of the system.Thus, for a realistic description of blood flow, perhaps, it is more appropriate to treat blood as a two-phase fluid model consisting of a core region inner phase containing all the erythrocytes as a nonNewtonian fluid and a peripheral layer outer phase of plasma as a Newtonian fluid.Several researchers have studied the two-phase fluid models for blood flow through stenosed arteries treating the fluid in the inner phase as a nonNewtonian fluid and the fluid in the outer phase as a Newtonian fluid 25, 26, 31-33 .Srivastava and Saxena 25 have analyzed a two-phase fluid model for blood flow through stenosed arteries treating the suspension of all the erythrocytes in the core region inner phase as a Casson fluid and the plasma in the peripheral layer outer phase is represented by a Newtonian fluid.In the present model, we study a two-phase fluid model for pulsatile flow of blood through stenosed narrow arteries assuming the fluid in the core region as a Herschel-Bulkley fluid while the fluid in the peripheral region is represented by a Newtonian fluid.
Chaturani and Ponnalagar Samy 28 and Sankar and Hemalatha 2 have mentioned that for tube diameter 0.095 mm blood behaves like Herschel-Bulkley fluid rather than power law and Bingham fluids.Iida  In this study, the effects of the pulsatility, stenosis, peripheral layer and the nonNewtonian behavior of blood are analyzed using an analytical solution.Section 2 formulates the problem mathematically and then nondimensionalises the governing equations and boundary conditions.In Section 3, the resulting nonlinear coupled implicit system of differential equations is solved using the perturbation method.The expressions for the velocity, flow rate, wall shear stress, plug core radius, and resistance to flow have been obtained.Section 4 analyses the variations of these flow quantities with stenosis height, yield stress, amplitude, power law index and pulsatile Reynolds number through graphs.The estimates of wall shear stress increase factor and the increase in resistance to flow factor are calculated for the two-phase Herschel-bulkley fluid model and single-phase fluid model.

Mathematical Formulation
Consider an axially symmetric, laminar, pulsatile and fully developed flow of blood assumed to be incompressible in the z direction through a circular artery with an axially symmetric mild stenosis.It is assumed that the walls of the artery are rigid and the blood is represented by a two-phase fluid model with an inner phase core region of suspension of all erythrocytes as a Herschel-Bulkley fluid and an outer phase peripheral layer of plasma as a Newtonian fluid.The geometry of the stenosis is shown in Figure 1.We have used the cylindrical polar coordinates r, φ, z whose origin is located on the vessel stenosed artery axis.It can be shown that the radial velocity is negligibly small and can be neglected for a low Reynolds number flow in a tube with mild stenosis.In this case, the basic momentum equations governing the flow are where the shear stress τ |τ r z | −τ r z since τ τ H or τ τ N .Herschel-Bulkley fluid is a nonNewtonian fluid which is widely used in many areas of fluid dynamics, for example, dam break flows, flow of polymers, blood, and semisolids.Herschel-Bulkley fluid is a nonNewtonian fluid with nonzero yield stress which is generally used in the studies of blood flow through narrow arteries at low-shear rate.Herschel-Bulkley equation is an empirical relation which connects shear stress and shear rate through the viscosity which is given in 2.4 and 2.5 .The relations between the shear stress and the strain rate of the fluids in motion in the core region for Herschel-Bulkley fluid and in the peripheral region for Newtonian fluid are given by where u H , u N are the axial component of the fluid's velocity in the core region and peripheral region; τ H , τ N are the shear stress of the Herschel-Bulkley fluid and Newtonian fluid; μ H , μ N are the viscosities of the Herschel-Bulkley fluid and Newtonian fluid with respective dimensions ML −1 T −2 n T and ML −1 T −1 ; ρ H , ρ N are the densities of the Herschel-Bulkley fluid and Newtonian fluid; p is the pressure, t; is the time; τ y is the yield stress.From 2.5 , it is clear that the velocity gradient vanishes in the region where the shear stress is less than the yield stress which implies a plug flow whenever τ H ≤ τ y .However, the fluid behavior is indicated whenever τ H ≥ τ y .The geometry of the stenosis in the peripheral region as shown in Figure 1 is given by where R z is the radius of the stenosed artery with peripheral layer, R 0 is the radius of the normal artery, L 0 is the length of the stenosis, d indicates its location, and δ p is the maximum depth of the stenosis in the peripheral layer such that δ P /R 0 1.The geometry of the stenosis in the core region as seen in Figure 1 is given by βR 0 in the normal artery region, where R 1 z is the radius of the stenosed core region of the artery, β is the ratio of the central core radius to the normal artery radius, βR 0 is the radius of the core region of the normal artery, and δ C is the maximum depth of the stenosis in the core region such that δ C /R 0 1.The boundary conditions are i τ H is finite and ∂u H ∂r

2.9
Since the pressure gradient is a function of z and t, we take where q z − ∂p/∂z z, 0 , f t 1 A sin ωt, A is the amplitude of the flow and ω is the angular frequency of the blood flow.Since any periodic function can be expanded in a series of sines of multiple angles using Fourier series, it is reasonable to choose f t 1 A sin ωt as a good approximation.We introduce the following nondimensional variables where , having the dimension as that of the Newtonian fluid's viscosity, q 0 is the negative of the pressure gradient in the normal artery, α H is the pulsatile Reynolds number or generalized Wormersly frequency parameter and when n 1, we get the Wormersly frequency parameter α N of the Newtonian fluid.Using the nondimensional variables, 2.1 , 2.2 , 2.4 , 2.5 , and 2.6 reduce, respectively, to 14 where f t 1 A sin t.The boundary conditions in dimensionless form are

2.17
The geometry of the stenosis in the peripheral region in dimensionless form is given by The geometry of the stenosis in the core region in dimensionless form is given by The nondimensional volume flow rate Q is given by where

Method of Solution
When we nondimensionalize the constitutive equations 2.1 and 2.2 , α 2 H and α 2 N occur naturally and these pulsatile Reynolds numbers are time dependent and hence, it is more appropriate to expand 2.12 -2.16 about α 2 H and α 2 N .The plug core velocity u p , the velocity in the core region u H , the velocity in the peripheral region u N , the plug core shear stress τ p , the shear stress in the core region τ H , the shear stress in the peripheral region τ N , and the plug core radius R p are expanded as follows in terms of α 2 H and α 2 N where α 2 H 1 and α 2 N 1 : Substituting 3.2 , 3.5 in 2.12 and then equating the constant terms and α 2 H terms, we obtain 3.9 Applying 3.2 , 3.5 in 2.14 and then equating the constant terms and α 2 H terms, one can get , ∂u 1P ∂r 0 at r 0, 3.17 u 1N 0 at r R z .

3.23
Equations 3.8 -3.11 and 3.12 -3.15 are the system of differential equations which can be solved for the unknowns u 0H , u 1H , τ 0H , τ 1H and u 0N , u 1N , τ 0N , τ 1N , respectively, with the help of boundary conditions 3.16 -3.23 .Integrating 3.8 between 0 and R 0P and applying the boundary condition 3.16 , we get τ 0P q z f t R 0P .

3.24
Integrating 3.8 between R 0P and r and then making use of 3.24 , we get τ 0H q z f t r.

3.25
Integrating 3.12 between R 1 and r and then using 3.18 , one can get τ 0N q z f t r.

3.26
Integrating 3.14 between r and R and then making use of 3.22 , we obtain

3.27
Integrating 3.10 between r and R 1 and using the boundary condition 3.20 , we get where k 2 θ/ q z f t .The plug core velocity u 0P can be obtained from 3.28 by replacing r by R 0P as

3.29
Neglecting the terms with α 2 H and higher powers of α H in 3.7 and using 3.24 , the expression for R 0P is obtained as

3.30
Similarly, solving 3.9 , 3.11 , 3.13 , and 3.15 with the help of 3.24 -3.29 , and using 3.19 , 3.21 and 3.23 , the expressions for τ 1P , τ 1H , τ 1N , u 1H , and u 1P can be obtained as where B 1/f t df t /dt .The expression for velocity u H can be easily obtained from 3.2 , 3.28 and 3.35 .Similarly, the expressions for u N , τ H , and τ N can be obtained.The expression for wall shear stress τ w can be obtained by evaluating τ N at r R and is given below:

3.37
From 2.20 and 3.27 , 3.28 , 3.29 , 3.34 , 3.35 , and 3.36 , the volume flow rate is calculated and is given by

Mathematical Problems in Engineering
The second approximation to plug core radius R 1P can be obtained by neglecting the terms with α 4 H and higher powers of α H in 3.7 in the following manner.The shear stress τ H τ 0H α 2 H τ 1H at r R P is given by

3.39
Equation 3.39 reflects the fact that on the boundary of the plug core region, the shear stress is the same as the yield stress.Using the Cityplace Taylor's series of τ 0H and τ 1H about R 0P and using τ 0H | r R 0P θ, we get With the help of 3.7 , 3.30 , 3.32 , and 3.40 , the expression for R P can be obtained as

3.41
The resistance to flow in the artery is given by Λ q z f t Q .

3.42
When R 1 R, the present model reduces to the single fluid model Herschel-Bulkley fluid model and in such case, the expressions obtained in the present model for velocity u H , shear stress τ H ,wall shear stress τ w , flow rate Q, and plug core radius R P are in good agreement with those of Sankar and Hemalatha 2 .

Numerical Simulation of Results and Discussion
The objective of the present model is to understand and bring out the salient features of the effects of the pulsatility of the flow, nonNewtonian nature of blood, peripheral layer and stenosis size on various flow quantities.It is generally observed that the typical value of the power law index n for blood flow models is taken to lie between 0.9 and 1.1 and we have used the typical value of n to be 0.95 for n < 1 and 1.05 for n > 1 2 .Since the value of yield stress is 0.04 dyne/cm 2 for blood at a haematocrit of 40 35 , the nonNewtonian effects are more pronounced as the yield stress value increases, in particular, when it flows through narrow blood vessels.In diseased state, the value of yield stress is quite high almost five times 28 .
In this study, we have used the range from 0.1 to 0.3 for the nondimensional yield stress θ.
To compare the present results with the earlier results, we have used the yield stress value as 0.01 0.04.Though the range of the amplitude A varies from 0 to 1, we use the range from 0.1 to 0.5 to pronounce its effect.
The ratio α α N /α H between the pulsatile Reynolds numbers of the Newtonian fluid and Herschel-Bulkley fluid is called pulsatile Reynolds number ratio.Though the pulsatile Reynolds number ratio α ranges from 0 to 1; it is appropriate to assume its value as 0.5 25 .Although the pulsatile Reynolds number α H of the Herschel-Bulkley fluid also ranges from 0 to 1 2 , the values 0.5 and 0.25 are used to analyze its effect on the flow quantities.Given the values of α and α H , the value of α N can be obtained from α α N /α H .The value of the ratio β of central core radius βR 0 to the normal artery radius R 0 in the unobstructed artery is generally taken as 0.95 and 0.985 25 .Following Shukla et al. 26 , we have used the relations R 1 βR and δ C βδ P to estimate R 1 and δ C .The maximum thickness of the stenosis in the peripheral region δ P is taken in the range from 0.1 to 0.15 25 .To compare the present results with the results of Sankar and Hemalatha 2 for single fluid model, we have used the value 0.2 for δ C .To deduce the present model to a single fluid model Newtonian fluid model or Herschel-Bulkley fluid model and to compare the results with earlier results, we have used the value of β as 1.
It is observed that in 3.38 , f t , R, and θ are known and Q and q z are the unknowns to be determined.A careful analysis of 3.38 reveals the fact that q z is the pressure gradient of the steady flow.Thus, if steady flow is assumed, then 3.38 can be solved for q z 2, 10 .For steady flow, 3.38 reduces to where x q z and Q S is the steady state flow rate.Equation 4.1 can be solved for x numerically for a given value of n, Q S and θ.Equation 4.1 has been solved numerically for x using Newton-Raphson method with variation in the axial direction and yield stress with β 0.95 and δ P 0.1.Throughout the analysis, the steady flow rate Q S value is taken as 1.0.Only that root which gives the realistic value for plug core radius has been considered there are only two real roots in the range from 0 to 20 and the other root gives values of plug core radius that exceeds the tube radius R .

Pressure Gradient
The variation of pressure gradient with axial distance for different fluid models in the core region is shown in Figure 2. It has been observed that the pressure gradient for the Newtonian fluid single fluid model is lower than that of the two fluid models with n 1.05 and θ 0.1 from z 4 to 4.5 and z 5.5 to 6, and higher than that of the two fluid models from z 4.5 to z 5.5 and these ranges are changed with increase in the value of the yield stress θ and a decrease in the value of the power law index n.The plot for the Newtonian fluid model single phase fluid model is in good agreement with that in Figure 2

Pressure Drop
The variation of pressure drop Δp across the stenosis, i.e., from z 4 to z 6 in a time cycle for different values of A, θ, and δ P with n β 0.95 is depicted in Figure 3.It is clear that the pressure drop increases as time t increases from 0 • to 90 • and then decreases from 90 • to 270 • and again it increases from 270 • to 360 • .The pressure drop is maximum at 90 • and minimum at 270 • .It is also observed that for a given value of A, the pressure drop increases with the increase of the stenosis height δ P or yield stress θ when the other parameters held constant.Further, it is noticed that as the amplitude A increases, the pressure drop increases when t lies between 0 • and 180 • and decreases when t lies between 180 • and 360 • while θ and δ P are held fixed.Figure 3 shows the simultaneous effects of the stenosis size and nonNewtonian nature of blood on pressure drop.

Plug Core Radius
The variation of plug core radius R P with axial distance for different values of the amplitude A and stenosis thickness δ P in the peripheral layer with n β 0.95, α H 0.5, θ 0.1, and t 60 • is shown in Figure 4.It is noted that the plug core radius decreases as the axial variable z varies from 4 to 5 and it increases as z varies from 5 to 6.It is further observed that for a given value of δ P , the plug core radius decreases with the increase of the amplitude A and the same behavior is noted as the peripheral layer stenosis thickness increases for a given value  of the amplitude A. Figure 4 depicts the effects of stenosis height on the plug core radius of the blood vessels.
Figure 5 sketches the variation of plug core radius in a time cycle for different values of the pulsatile Reynolds number α H of the Herschel-Bulkley fluid and yield stress θ with n β 0.95, A 0.5, z 5, t 60 • , and δ P 0.1.It is noted that the plug core radius decreases as time t increases from 0 • to 90 • and then it increases from 90 • to 270 • and then again it decreases from 270 • to 360 • .The plug core radius is minimum at t 90 • and maximum at t 270 • .It has been observed that for a given value of the pulsatile Reynolds number α H , the plug core radius increases as the yield stress θ increases.Also, it is noticed that for a given value of yield stress θ and with increasing values of the pulsatile Reynolds number α H , the plug core radius increases when t lies between 0 • and 90 • and also between 270 • and 360 • and decreases when t lies between 90 • and 270 • .Figure 5 depicts the simultaneous effects of the pulsatility of the flow and the nonNewtonian nature of the blood on the plug core radius of the two-phase model.

Wall Shear Stress
Wall shear stress is an important parameter in the studies of the blood flow through arterial stenosis.Accurate predictions of wall shear stress distributions are particularly useful in the Mathematical Problems in Engineering   understanding of the effects of blood flow on the endothelial cells 36, 37 .The variation of wall shear stress in the axial direction for different values of yield stress θ and pulsatile Reynolds number α N of the Newtonian fluid with t 45 • , n β 0.95, A 0.5, and δ P 0.1 is plotted in Figure 6.It is found that the wall shear stress increases as the axial variable z increases from 4 to 5 and then it decreases symmetrically as z increases further from 5 to 6.For a given value of the pulsatile Reynolds number α N , the wall shear stress increases considerably with the increase in the values of the yield stress θ when the other parameters held constant.Also, it is noticed that for a given value of the yield stress θ and increasing values of the pulsatile Reynolds number α N , the wall shear stress decreases slightly while the other parameters are kept as invariables.It is of interest to note that the plot for the single fluid Herschel-Bulkley model is in good agreement with that in Figure 8 of Sankar and Hemalatha 2 .Figure 6 shows the effects of pulsatility of the blood flow and nonNewtonian effects of the blood on the wall shear stress of the two-phase model.
Figure 7 depicts the variation of wall shear stress in a time cycle for different values of the amplitude A and peripheral stenosis height δ P with n β 0.95, θ 0.1, α N 0.5 and z 5.It can be easily seen that the wall shear stress increases as time t in degrees increases from 0 • to 90 • and then it decreases as t increases from 90 • to 270 • and then again it increases as t increases further from 270 • to 360 • .The wall shear stress is maximum at 90 • and minimum at 270 • .Also, it may be noted that for a given value of the amplitude A the wall shear stress increases with increasing values of the stenosis thickness δ P .Further, it is noticed that for a given value of the stenosis size and increasing values of the amplitude A, the wall shear stress increases when t lies between 0 • and 180 • and decreases when t lies between 180 • and 360 • .This figure sketches the effects of the stenosis size and amplitude on the wall shear stress of the two-phase blood flow model.

Velocity Distribution
The velocity profiles are of interest, since they provide a detailed description of the flow field.The velocity distributions in the radial direction for different values of the amplitude A, pulsatile Reynolds number ratio α, pulsatile Reynolds number of Herschel-Bulkley fluid α H , the ratio of the central core radius to the tube radius β with n 0.95, z 5, θ δ P 0.1, and t 45 • are shown in Figure 8.One can easily notice the plug flow around the tube axis in Figure 8. Also, it is found that the velocity increases as the amplitude A increases for a given set of values of α, α H and β.Further, it is observed that for a given set of values of A, α and α H , the velocity decreases considerably near the tube axis as the ratio β increases.The same behavior is observed for increasing values of the pulsatile Reynolds number ratio α and pulsatile Reynolds number α H for the given values of A and β, but there is only a slight decrease in the later case.

Resistance to Flow
The variation of resistance to flow with peripheral layer stenosis size for different values of the amplitude A and yield stress θ with n β 0.95, α α H 0.25, and t 45 • is plotted in Figure 10.Since δ C βδ P , the stenosis size of the core region δ C also increases when the     peripheral layer stenosis height δ P increases for a given value of β.It is seen that the resistance to flow increases gradually with increasing stenosis size while the rest of the parameters are kept fixed.It is to be noted that for a given value of yield stress θ, the resistance to flow decreases with increasing values of the amplitude A. It is also found that for a given value of the amplitude A, the resistance to flow increases with increase in the values of the yield stress θ when the other parameters held constant.Figure 10 illustrates the effects of the amplitude, stenosis size and the nonNewtonian nature of blood on resistance to flow of the two-phase model.
Figure 11 sketches the variation of resistance to flow in a time cycle for different values of the power law index n and the pulsatile Reynolds number ratio α, pulsatile Reynolds number of the Herschel-Bulkley fluid α H with θ δ P 0.1, β 0.95 and A 0.2.It is clear that the resistance to flow decreases as time t in degrees increases from 0 • to 90 • and then it increases as t increases from 90 • to 270 • and then again it decreases as t increases further from 270 • to 360 • .The resistance to flow is minimum at 90 • and maximum at 270 • .It is found that for the fixed values of α and α H and the increasing values of the power law index n, the resistance to flow decreases when time t lies between 0 • and 180 • and increases when t lies between 180 • and 360 • .Further, it is noted that for a fixed value of the power law index n and with the increasing values of α and α H , the resistance to flow increases slightly when t Problems in Engineering  lies between 0 • and 90 • and also between 270 • and 360 • and decreases slightly when t lies between 90 • and 270 • .Figure 11 shows the simultaneous effects of pulsatility of the flow and the nonNewtonian nature of blood on resistance to flow of the two-phase model.

Quantification of Wall Shear Stress and Resistance to Flow
The wall shear stress τ w and resistance to flow Λ are physiologically important quantities which play an important role in the formation of platelets 38 .High wall shear stress not only damages the vessel wall and causes intimal thickening, but also activates platelets, cause platelet aggregation, and finally results in the formation of thrombus 7 .
The wall shear stress increase factor is defined as the ratio of the wall shear stress of particular fluid model in the stenosed artery for a given set of values of the parameters to the wall shear stress of the same fluid model in the normal artery for the same set of values of the parameters.The estimates of the wall shear stress increase factor for twophase Herschel-Bulkley fluid model and single-phase fluid model with t 45 • , n 0.95, A α α H 0.5, β 0.985, and θ 0.1 are given in Table 1.It is observed that for the range of the stenosis size 0-0.15, the corresponding ranges of the wall shear stress increase of the two-phase Herschel-Bulkley fluid model and single-phase Herschel-Bulkley fluid model are 1.074-1.594and 1.156-2.848,respectively.It is found that the estimates of the wall shear stress increase factor are marginally lower for the two-phase Herschel-Bulkley fluid model than those of the single-phase Herschel-Bulkley fluid model.One can define the resistance to flow increase factor in a similar way as in the case of wall shear stress increase factor.The estimates of the increase in resistance to flow factor for two-phase Herschel-Bulkley fluid model and single-phase fluid model with t 45 • , n 0.95, A α α H 0.5, β 0.985, and θ 0.1 are given in Table 2.It is noted that for the range of the stenosis size 0-0.15, the corresponding range of the increase in resistance to flow factor for the two-phase Herschel-Bulkley fluid model and single-phase Herschel-Bulkley fluid model are 1.050-1.393and 1.104-2.189,respectively.It is found that the estimates of the wall shear stress increase factor are significantly lower for the two-phase Herschel-Bulkley fluid model than those of the single-phase Herschel-Bulkley fluid model.Hence, it is clear that the existence of the peripheral layer is useful in the functioning of the diseased arterial system.It is strongly felt that the present model may provide a better insight to the study of blood flow behavior in the stenosed arteries than the earlier models.
Perturbation method is a very useful analytical tool for solving nonlinear differential equations.In the present study, it is used to solve the nonlinear coupled implicit system of partial differential equations to get an asymptotic solution.This method yields a closed form to the flow quantities which enables us to evaluate them at any particular instant of time and at any particular point in the flow domain.This facility is unavailable when we use the computational methods such as finite difference method, finite element method, finite volume method.

Conclusion
The present study analyzes the two-phase Herschel-Bulkley fluid model for blood flow through stenosed arteries and brings out many interesting fluid mechanical phenomena due to the presence of the peripheral layer.The results indicate that the pressure drop, plug core radius, wall shear stress, and resistance to flow increase as the yield stress or stenosis size increases while all other parameters held constant.It is found that the velocity increases, plug core radius, and resistance to flow decrease as the amplitude increases.It is also observed that the difference between the estimates of increase in the wall shear stress factor of the two-phase fluid model and single-phase fluid model is substantial.A similar behavior is observed for the increase in resistance to flow factor.Thus, the results demonstrate that this model is capable of predicting the hemodynamic features most interesting to physiologists.Thus, the present study could be useful for analyzing the blood flow in the diseased state.From this study, it is concluded that the presence of the peripheral layer outer phase helps in the functioning of the diseased arterial system.amplitude of the flow q z : steady state pressure gradient q z : dimensionless steady state pressure gradient q 0 : negative of the pressure gradient in the normal artery L:

LFigure 1 :
Figure 1: Flow geometry of an arterial stenosis with peripheral layer.

fluid model with β 1 Figure 2 : 1 A 1 A 1 Figure 3 :
Figure 2: Variation of pressure gradient with axial direction for different fluids in the core region with δ P 0.1.

Figure 5 :
Figure 5: Variation of plug core radius in a time cycle for different values of α H and θ with n β 0.95, δ P 0.1, A 0.5, t 60 • and z 5.

1 θFigure 6 :
Figure 6: Variation of wall shear stress with axial distance for different values θ and α N with t 45 • , n β 0.95, A 0.5 and δ P 0.1.

Figure 7 :
Figure 7: Variation of wall shear stress in a time cycle for different values of A and δ P with θ 0.1, n β 0.95, z 5 and α N 0.5.
Figure 8 depicts the effects of amplitude, pulsatility and stenosis size on velocity distribution of the two-phase model.The velocity distribution in the radial direction at different times is shown in Figure 9.It is observed that the velocity increases as time t in degrees increases from 0 • to 90 • and then it decreases as t increases from 90 • to 270 • and again it increases as t increases further from 270 • to 360 • .This figure shows the transient effects of blood flow on velocity of the two-phase model.

1 Figure 10 :
Figure 10: Variation of resistance to flow with stenosis size for different values of A and θ with n β 0.95, α α H 0.25 and t 45 • .

2 n 2 Figure 11 :
Figure 11: Variation of resistance to flow in a time cycle for different values of α, α H and n with θ δ P 0.1, β 0.95, and A 0.2.
R 0 : radius of the normal artery R z : radius of the artery in the stenosed peripheral region R z : dimensionless radius of the artery in the stenosed peripheral region R 1 z : radius of the artery in the stenosed core region R 1 z : dimensionless radius of the artery in the stenosed core region R P : plug core radius R P : dimensionless plug core radius u H : axial velocity of the Herschel-Bulkley fluid u H : dimensionless axial velocity of the Herschel-Bulkley fluid u N : axial velocity of the Newtonian fluid u N : dimensionless axial velocity of the Newtonian fluid A:

Δp:
dimensionless Pressure drop Λ: dimensionless resistance to flow φ: azimuthal angle γ: shear rate τ y : yield stress θ: dimensionless yield stress τ H : shear stress for the Herschel-Bulkley fluid τ H : dimensionless shear stress for the Herschel-Bulkley fluid Mathematical Problems in τ N : shear stress for the Newtonian fluid τ N : dimensionless shear stress for the Newtonian fluid τ w : dimensionless wall shear stress ρ H : density of the Herschel-Bulkley fluid ρ N : density of the Newtonian fluid μ H : viscosity of the Herschel-Bulkley fluid μ N : viscosity of the Newtonian fluid α H : pulsatile Reynolds number of the Herschel-Bulkley fluid α N : pulsatile Reynolds number of the Newtonian fluid α: ratio between the Reynolds numbers α H and α N β: ratio of the central core radius to the normal artery radius δ C : maximum height of the stenosis in the core region δ C : dimensionless maximum height of the stenosis in the core region δ N : maximum height of the stenosis in the peripheral region δ P : dimensionless maximum height of the stenosis in the peripheral region ω: angular frequency of the blood flow.

Table 1 :
Estimates of the wall shear stress increase factor for the two-phase Herschel-Bulkley fluid model and single-phase Herschel-Bulkley fluid model for different stenosis sizes with n 0.95, A α α H 0.5, β 0.985, θ 0.1, and t 45 • .

Table 2 :
Estimates of the resistance to flow increase factor for the two-phase Herschel-Bulkley fluid model and single-phase Herschel-Bulkley fluid model for different stenosis sizes with n 0.95, A α α H 0.5, β 0.985, θ 0.1, and t 45 • .